Sound,

In the 18th century it became clear that sound was not some particles flying from a source to the listener, the string cutting up the air into little pieces which went flying away. Instead sound was a series of small pressure variations in the air which travelled away from the source as waves. If two waves travel through each other, the net pressure is just the sum of the two pressure variations.

Beats

If one has two sounds with almost the same frequency, instead of hearing two separate sounds one hears a single pitch which gets louder and quieter at a rate given by the difference of the two frequencies.

Hamonics

When one plays a note on any instrument, it is not just that one sound which is generated, but a series of harmonics, which happen to be integer multiples of the original note. A sinusoidal pressure variation is a "pure note" with no harmonics except itself. Any normal note is made up of a sequence of such sinusoidal pitches which give the note its richness. For any continous note instrument (bowed string, blown trumpet, or clarinet, organ pipe,...) each note is accompanied by these harmonics, each of which is an interger multiple of the bottom frequency.

The astonishing feature of the "ear" (meaning the whole process of hearing from the sound entering into the ear drum to the brain intereting those sounds) is that this whole suite of sounds is melded together into a single note, a single pitch somehow.

the Ear

The ear has three parts-- the outer ear ( the visible ear) whose main purpose is to funnel the sound into the inner ear, which starts with the ear drum, which the pressures in the sound wave move back and forth.

The ear drum's motion pushes back and forth on a series of little bones which finally push and pull on a fluid filled chamber, called the cochlea. That chamber is a long (about 3 cm) tapered chamber which is divided into two longitudally but a couple of thin membranes highly populated by nerve cells. These nerve cells respond to those memberanes being bent, and convey that information to the brain. This membrane is tuned, so that if a sound of a certain frequency comes into the innter ear, one a tine piece (less than a millimeter in length) vibrates to each specific frequency in the sound. Thus each of the harmonics of a note produce motion in different parts of the basilar membrane, and the nerves send to the brain information from each of those parts separately. Thus the brain receives, not something which is a replica of the sound itself, but rather separately with respect to each of the harmonics of the sounds coming in.

Thus one would think that the brain would report on all of those frequencies, and one would hear any sound, not as a single pitch, but as a whole cacophany of sounds. But that is not what happens. Instead the brain somehow reassembles the sounds into a single pitch experience, in which, if you listen really carefully, and with training, one can just about hear a few of separate sounds.

It does so even in the presence of other sounds. Thus a violin note does not split up into its separate harmonics if one has someone talking in the foreground or even has other instruments playing.

If one plays two notes a fifth apart (frequency ratio of 3/2) then the the two notes together tend to merge together. The frequencies in the combination has frequencies 1 3/2 2 3 3 (4 1/2) 5 6 6 .. Ie one has Ie, one again has a regular series of frequencies, which the brain tends to regard as a series which it should meld together into one experience.

The fourth has a similar series
1 4/3 2 8/3 3 4 4 5 (5 1/3) 6 (6 2/3) 7 ...
which are all multiples of 1/3. Again, this regular sequence somehow makes the brain think that it should unify all those notes into one experience. Similarly for a major third, and a major sizth.

If one happens to have a fifth which is slightly off tune, then one will get beats between the harmonics which the two series have in common. (The double 3 or double 6 will beat with each other. ) These beats will make the sound of the two notes together feel non-pefect. The brain is trying to meld the notes together while the beats get in the way of that. While the ear is willing to forgive some disharmony in the two note together, it gets less and less willing to the more harmonics the two notes share. Thus the octave, in which all of the harmonics of the higher note are identical to harmonics of the lower note, the ear is particularly intolerant of a mistuning. The fifth is next, the fourth is next, then the third and the sixth, etc.

Pythagoras and harmony

Pythagoras' discovery of the harmony of notes which has a small whole number ratio in what we know was their frequencies (he saw them as ratios of lengths of the string that was vibrating) was the begining of the mathematisation of the physical world. He was struck by the fact that there was some relationship between mathematics, a product of the human imagination, and the outside physical world. That such a relation was possible was a shock and a goad to human understanding.

It is ironic that even now, why the ear is so frequency sensitive and at the same time, the brain melds the harmonics into individual pitches, is still somewhat of a mystery. We do not understand why and how the human mind melds together certain sounds into one single exerience, instead of hearing it as a cacophany. But along the way, that insight of Pythagoras resulted in a steady worry in exactly the people who were most important to our current scientific understanding of the world. And it led to the discovery of what sound is, of the prevelance of harmonics hidden in each note that we hear, and in the whole study of how the brain interacts with the physical world.

One of the key features of the human hearing system is that it is the most sensitive of the senses we have. A trained human can differentiate between sounds which are closer than one part in 10000 of each other and does so over about 10 octaves (another factor of a thousand.) Thus we can differentiate between about 1 to 10 million different sounds just in terms of frequency. At the same time, the brain also melds the harmonics into different tambres (so that a clarinet and a violin sound different, even though both play the same pitch. For example the eye is sensitive to less than a factor of 1/2 in frequencies. It also is sensitive to differences in tambre (colours in the case of the eye) but the total sensitivity is much less than that of the ear.